- either a graduate level one semester course in partial differential equations, a theoretical graduate level course in an applied field such as fluid mechanics, or permission of instructor.

- Classical Mechanics
- Many-Body Problem
- Lagrangian Formulation
- Hamiltonian Formulation
- Rigid-Body Dynamics
- Vibrating-Body Dynamics

- Fluid Dynamics
- Local Conservation and Balance Laws
- Compressible Euler Systems
- Hyperbolicity
- Equilibrium Thermodynamics and Entropy
- Compressible Navier-Stokes Systems

- Incompressible Fluid Dynamics
- Incompressible Regimes
- Boussinesq-Balance Stokes, Navier-Stokes, and Euler Systems
- Dominant-Balance Stokes, Navier-Stokes, and Euler Systems

- Kinetic Theories
- Microscopic Pictures
- Kinetic Regimes
- Collisionless Regimes
- Collisional Regimes
- The Boltzmann-Grad Limit

- Maxwell-Boltzmann Theory
- The Classical Boltzmann Equation
- Boltzmann Collision Operators
- Elastic Binary Collisions
- Gain and Loss Terms
- Collision Kernels
- Boundary Conditions

- Properties of the Boltzmann Equation
- Galilean Symmetry
- Boltzmann Identities
- Collision Invariants
- Local Conservation
- Entropy, Local Dissipation, and Equilibria
- The Maxwell Continuity Equation

- Recipes for Collision Kernels
- Maxwell's Recipe for Classical Monatomic Molecules
- Cut-off Kernels
- Fermi's Recipe for Quantum Molecules

- The Classical Boltzmann Equation
- Classical Kinetic Theories
- Properties of Collision Operators
- Galilean Symmetry
- Local Conservation
- Entropy, Local Dissipation, and Equilibria

- Example: Born-Green-Kirkwook Operators
- Example: Fokker-Planck-Landau Operators
- Example: Linear and Linearized Theories

- Properties of Collision Operators
- More General Kinetic Theories
- Properties of Collision Operators
- Local Conservation
- Entropy, Local Dissipation, and Equilibria

- Example: Fermi-Dirac and Bose-Einstein Theories
- Example: Polyatomic Models
- Example: Multispecies Models
- Example: Discrete Velocity Models
- Example: Linear and Linearized Theories
- Properties of Boundary Operators

- Properties of Collision Operators

- Compressible Euler Limits
- Fluid Dynamical Regimes
- The Knudsen Number
- Classical Compressible Euler Limits
- General Compressible Euler Limits

- General Compressible Euler Systems
- Potential Formulation
- Density Formulation
- Characteristic Velocities

- Fluid Dynamical Regimes
- Beyond Compressible Euler Limits
- The Hilbert Approach
- Hilbert Expansions
- Assumptions on the Linearized Collision Operator
- Order-by-Order Solution
- The Navier-Stokes Approximation
- The Hilbert-Grad Theory for Cut-Off Hard Kernels
- The Golse-Poupaud Theory for Cut-Off Soft Kernels

- The Chapman-Enskog Approach
- Chapman-Enskog Equations
- Chapman-Enskog Expansions
- Assumptions on the Linearized Collision Operator
- Order-by-Order Solution
- The Navier-Stokes Approximation
- Problems with Formal Well-Posedness at Higher Order

- Self-Consistency of Classical Fluid Systems
- Classical Fluid Systems and Moments
- Moment Realizability
- Moment-Based Self-Consistency Criteria

- Moment Systems
- The Moment Closure Problem
- Grad Closures
- Entropy-Based Closures
- Recovering Fluid Dynamics

- The Hilbert Approach
- Linear and Weakly Nonlinear Regimes
- General Setting
- Nondimensional Form
- Conservation and Dissipation Laws
- Fluctuations
- Moment-Based Derivations

- Derivation of Acoustic Systems
- Infinitesimal Maxwellians
- The Acoustic System

- Derivation of Boussinesq-Balance Systems
- Long-Time Scaling
- The Incompressibility and Boussinesq Relations
- The von Karman Relation
- The Key Ideas of the Derivation
- Assumptions on the Linearized Collision Operator
- The Stokes, Navier-Stokes, and Euler Dynamics

- Derivation of Dominant-Balance Systems
- The Odd-Even Splitting
- The Incompressibility and Pressure Relations
- The Key Ideas of the Derivation
- Assumptions on the Collision Operator
- Identities for the Collision Operator
- The Stokes, Navier-Stokes, and Euler Dynamics

- Derivation of Thermal-Stress Systems
- Thermal-Stress Regimes
- The Incompressibility and Pressure Relations
- The Key Ideas of the Derivation
- Assumptions on the Collision Operator
- The Dynamics

- General Setting

- Global Existence and Uniqueness Theories for Fluids
- Well-Posedness of the Acoustic and Stokes Equations
- Leray Existence Theory for the Incompressible Navier-Stokes Equations
- Leray Uniqueness Theory for Classical Solutions of the Incompressible Navier-Stokes Equations
- Dissipative Solutions of the Incompressible Euler Equations

- Regularity Theories for Fluids
- Regularity of Solutions of the Incompressible Euler Equations
- Regularity of Solutions of the Incompressible Navier-Stokes Equations

- Analytical Tools for Kinetic Equations
- Characteristics, $L^p$ and $L^\Phi$ Estimates for the Transport Equation
- Compactness in the Weak $L^1$ Topology
- Velocity Averaging
- Example: Parabolic Scalings and the Rosseland Approximation
- More Velocity Averaging
- Example: Hyperbolic Scalings and the Perthame-Tadmor BGK model

- Global Existence and Uniqueness Theories for Kinetic Models
- Space Homogeneous Solutions of the Boltzmann Equation
- Kaniel-Shinbrot Near Vacuum Theory
- Near Equilibrium Theory: $L^2$ Theory of the Linearized Boltzmann Equation (Grad)
- Global Existence for the BGK Model

- DiPerna-Lions Theory
- Renormalized Solution of the Boltzmann Equation (DiPerna-Lions)
- Regularity of the Gain Term
- Propagation of Singularities for the Boltzmann Equation
- Proof of the DiPerna-Lions Theorem
- Extensions the DiPerna-Lions Result
- Uniqueness of Classical Solutions

- Local Hydrodynamic Limits for Smooth Solutions
- Results Based on the Hilbert Expansion
- Results Based on the Cauchy-Kovalevska Theorem
- The Bardos-Ukai Theorem on the Incompressible Navier-Stokes Limit for Small Initial Data

- Global Hydrodynamic Limits: the BGL Program
- Fluctuation Theory for Renormalized Solutions
- The Linearized Boltzmann Limit
- The Acoustic Limit
- An Incompressible Stokes-Fourier Limit
- More Fluctuation Theory
- An Incompressible Navier-Stokes-Fourier Limit

- The Relative Entropy Method
- Evolution of the Relative Entropy: Formal Computations
- An Incompressible Euler Limit
- The Compressible Stokes Limit